Time-Efficient Quasi-Static Algorithm for Simulation of Complex Single-Sided Clamped Electrostatic Actuators

JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 16, NO. 2, APRIL 2007 373

Time-Efficient Quasi-Static Algorithm forSimulation of Complex Single-Sided

Clamped Electrostatic ActuatorsJoachim Oberhammer, Member, IEEE, A.-Q. Liu, and Göran Stemme, Fellow, IEEE

Abstract—This paper reports on a very time- and resource-efficient numerical algorithm for quasi-static modeling of the staticbehavior and the “quasi-static movement” of highly nonlinearelectrostatic actuators with single-side clamped moving elements.The algorithm is capable of simulating prestressed materialsand multicontact touching surfaces with complex geometries,including distance-keeping stoppers and thickness and materialinhomogeneities of the moving parts. Thus, it is very suitablefor predicting the behavior of actuators such as laterally movingcurved-electrode actuators or vertically moving touch-mode orzipper actuators. In contrast to conventional, very time- andmemory-consuming simulation methods such as finite-elementanalysis, the proposed algorithm—even if implemented in the slowscript-language of MATLAB—takes only a fraction of a second tosolve a complex problem, which makes it a very powerful designtool for parameter optimization of the actuator geometry. Thereason for the efficiency of this algorithm is that its core is basedon the one-dimensional mathematical description of a two-di-mensional model geometry and that the differential equationis solved by a simple triple-integration for each iteration step,which is a method very suitable for thin-film single-side clampedmoving elements. This paper describes the algorithm, analyzes itsaccuracy and its limitations, and reports on its performance ascompared to other methods such as simplified analytical modelsfor very basic structures, finite-element method (FEM) simu-lations of complex structures, and measurements of fabricateddevices, including laterally moving microelectromechanical sys-tems (MEMS) switches and vertically closing prestressed thin-filmzipper actuators. Furthermore, the efficiency of the algorithmas a design tool was evaluated for the parameter optimization ofelectrostatic curved-electrode actuators. The algorithm’s mainapplication is seen in the fast determination of suitable parametersets for MEMS electrostatic actuators, but it cannot substitute fora more accurate FEM analysis to investigate a final design in greatdetail. [1725]

Index Terms—Curved-electrode actuator, design optimization,electrostatic actuator, microelectromechanical systems (MEMS)design, MEMS simulation, quasi-static modeling, touch-modeactuator.

Manuscript received November 25, 2005; revised September 1, 2006. SubjectEditor N. Aluru.

J. Oberhammer and G. Stemme are with the Microsystem Technology Group,School of Electrical Engineering, Royal Institute of Technology, SE-100 44Stockholm, Sweden.

A.-Q. Liu is with Photonics Lab II, School of Electrical and Electronics En-gineering, Nanyang Technological University, 639798 Singapore.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2007.892917

I. INTRODUCTION

ELECTROSTATIC actuation is a concept of high interestfor generating moving microsystems because of the high-

energy densities and large forces due to the scaling laws insmall dimensions and because of relatively simple fabrication[1]. However, the main disadvantage of electrostatic actuationis the limitation to small displacement. For practical designs(medium actuation voltage, electrode area, and reliable restoringspring force) of parallel-plate electrode arrangements and move-ment in the direction of the field lines, the maximum travelingdistance of the moving electrode is limited in most practical de-signs to a few micrometers.

Curved-electrode actuators, also known as touch-mode,moving-wedge, or rolling-contact actuators, are electrostaticactuators with single-side clamped moving structures and arebased on an electrode geometry with the electrode distancegradually increasing from the clamped end to the free end ofthe moving structure. Due to the narrow gap in the beginning,high forces initialize the movement of successive parts of theflexible electrode and the short electrode distance, and thusthe location of the large actuation force is moving along thefixed electrode in a zipper-like way [2], [3]. Such an actuatorachieves very large deflection of the free end of the movingstructure at relatively low actuation voltages. Another featureof electrostatic curved-electrode actuators, in contrast to elec-trothermal actuators that achieve a similar large displacement,is that the maximum force is created in the end-position ofthe movement, which makes them very suitable for electricalmicroswitches.

The two most common types of such zipper actuators area) vertically moving touch-mode actuators [2], as shown inFig. 1(a), where the prebending of the moving part is usuallycreated by a controlled, fabrication process–related stress gra-dient, and b) laterally moving curved-electrode actuators [4],as shown in Fig. 1(b), where the shape of the curved electrodeis simply defined by a photolithographical mask whose patternis vertically etched into the device layer. Among others, theseactuators have found applications in MEMS electrical switches[5]–[8], tunable capacitors [9], [10], gas valves [11]–[14], andfor microoptical applications [15], [16].

However, the increased academic and industrial research anddevelopment activities on such actuators lack good and fast de-sign tools. The main requirements on such a design tool are topredict the static and dynamic behavior of the actuator and toallow for the optimization of the actuator parameters, such as

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374 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 16, NO. 2, APRIL 2007

Fig. 1. The two most common implementations of zipper actuators: (a) ver-tically moving touch-mode actuator: flexible, initially bent curved electrode,pulled-in toward the fixed flat counterelectrode; (b) laterally moving curved-electrode actuator: flexible, initially straight moving electrode bending alongthe curved, fixed electrode.

the geometry and the actuation voltage. The complexity of thetype of actuators under discussion evolves many problems forthe simulation model, mainly because of

• the highly nonlinear nature of the deflection-dependentforces (requiring a nonlinear solver);

• the large change in the electrode distance (requiring anadaptive mesh and moving boundaries);

• touching surfaces for the zipping or rolling motion (mul-tiple contact problem);

• material prestress determining the initial bending of themoving structure, especially for vertically moving zipperactuators.

Analytical solutions of nonparallel electrode problemsare not available. Only for very simple structures describ-able by analytical functions, the energy equilibrium at theinstable snap-in point can be described by a nonlinear dif-ferential equation [1]. However, an analytical closed-formsolution of that differential equation cannot be found, but itis possible to construct an approximate solution by a simpli-fied model based on the Rayleigh–Ritz method [17], whichallows at least for an estimation of the necessary pull-involtage [1]. Finite-element/boundary-element (FEM/BEM),finite-cloud/boundary-cloud (FCM/BCM), or mixed-domainanalysis-based simulation software [18]–[20] can predict thehighly nonlinear behavior of touching structures and complexgeometries but require complicated coupled electrostatic-me-chanical domain models with moving boundaries and thecapability of solving multiple-contact problems. Furthermore,because of the large ratio between lateral and vertical di-mensions of typical MEMS thin-film structures, the mesh ofaccurate FEM models has to consist of a large number ofelements with an exploding number of degrees of freedom(DOFs), which results in memory-consuming matrices to besolved. Simulating such models is very time-consuming andoften results in convergence problems. Therefore, they are lesssuitable for the optimization of design parameters such as theelectrode shape, the cantilever geometry, or the stopper sizesand positions.

The presented numerical algorithm allows for quasi-staticsimulation of the static basic dynamic behavior of two-di-mensional single-side clamped structures and is capableof handling complex cantilever and electrode geometriesincluding stoppers, isolation layers, multilayer cantilever struc-tures, inhomogeneous material properties, and prestressedmaterials, which typically determine the curvature of vertically

Fig. 2. Geometrical model of the actuator with isolation layers and stoppersinternally represented as small isolation layer bumps.

closing zipper actuators. The algorithm is faster by a few ordersof magnitude than finite-element methods and therefore verysuitable for design parameter optimization.

II. DESCRIPTION OF THE ALGORITHM

A. Geometrical Model

The algorithm is based on a two-dimensional geometrical de-scription of the two-electrode problem. The deflecting struc-ture is mathematically modeled one-dimensionally and thus as-sumed to be very thin compared to its length. This descriptionis appropriate for typical single-side clamped electrostatic actu-ators because the cantilever thickness is typically very small ascompared to its length. The geometry of the fixed electrode aswell as the geometry of an optional isolation layer on that elec-trode are described by numerical vectors and thus can have a dis-continuous shape, which allows for complex electrode featuressuch as stoppers and isolation layers of any shape, as shown inFig. 2. Geometrical discontinuities and multiple layers of theflexible structure are modeled by variable material parametersover the cantilever length (Young’s modulus and moment ofinertia ). The initial curvature of the flexible structure under anoptional material prestress is modeled by a distributed torque

or by an external force distribution additionallyacting on the structure.

A two-dimensional model is sufficient to describe mostMEMS actuators of this type. The depth of the electrodes inthe -axis does not affect the behavior of the actuator, since theelectrostatic force is linearly increasing with the electrode depthand therefore compensating for the also linearly increasingcounteracting mechanical spring force (neglecting fringe-fieldeffects).

Multilayer cantilevers, i.e., structures with a cross-sec-tion composed by a stack of homogeneous materials such asbimetallic strips, e.g., can be simulated by using the materialparameters of an equivalent cross-section, i.e., an effective

-product representing the equivalent stiffness of the beam.The equivalent stiffness can be calculated by formulasfrom textbooks [21, Section 8.2] or directly be derived from itsdefinition

(1)

with the total cantilever thickness and the width of the struc-ture in the third dimension.

OBERHAMMER et al.: SIMULATION OF COMPLEX SINGLE-SIDED CLAMPED ELECTROSTATIC ACTUATORS 375

Fig. 3. Diagram illustrating how the forces and torques are calculated for asingle iteration step of the space discretizised model (for simplicity reasons,without external forces and material prestress): (a) the electrodes are modeledby distributed discrete capacitors; (b) the forces between the plates of each ca-pacitor are calculated; and (c) the torque at each single discretization point xis calculated by summing up all of the torques created by the forces on the sidetoward the free end in relation to the point x .

B. Quasi-Static Iteration Solver

The two main steps of a single quasi-static iteration step ofthe electrostatic/elastomechanical coupled-domain analysis are[22, Section 7.3.6]:

• electrostatic domain: determine the electrostatic forces im-posed on the current geometry;

• elastomechanical domain: determine the deformation (new geometry) due to these forces.

The elastomechanical description of the deflection in a singleiteration step is based on the classical Euler–Bernoulli beamdifferential equation [23]

(2)

describing the local curvature of a structure with the moment ofinertia and Young’s modulus bending due to the torquedistribution . This equation is valid for “small” deflectionsof infinitesimal thin structures. The sum of moments ineach position is numerically derived from the electrostaticforce distribution , as shown in Fig. 3, and optionally in-fluenced by mechanical prestress and an external forcedistribution

(3)

or for the space-discretized model

(4)

The electrostatic force is the attracting force betweenthe plates of—for the space-continuous model infinites-imal small—parallel-plate capacitors distributed along theelectrodes. For the space-discretized model, the number ofcapacitors is given by the number of nodes on the cantileverand the width of each capacitor is given by the cantilever length

divided by . The distance between the two electrodes ofeach capacitor is determined by the cantilever’s deflection of

the previous iteration step, i.e., the electrostatic force of a singleelement is calculated by

(5)

with the width of the cantilever and thus the electrodes inthe third dimension, the actuation voltage, and thelocal thickness and dielectric constant of the dielectric layer, ifexisting, the description of the curved electrode’s shape,and the deflection of the cantilever calculated during the lastiteration step at the position .

The deflection of the flexible structure is calculated by doublenumerical integration of (2), which is for the space-discretizedmodel

(6)

with from (4) and the moment of inertia optionally vari-able over the cantilever length. and are the integrationconstants representing the -offset and initial slope of the can-tilever, which are especially necessary to be taken into accountwhen continuing the simulation after a contact has occurred (seeSection II-C). The new deflection is used for calculating thenew electrostatic force distribution in the subsequent iterationstep.

Thus, the two main mathematical steps of the algorithm forsolving a single iteration are the calculation of the electrostaticforce distribution and the triple-integration thereof according to(2) and (3). This straightforward approach without solving anynonlinear differential equation systems is the reason for the ex-tremely short simulation times and the negligible memory con-sumption of the presented algorithm, with all its limitations andinaccuracies resulting from the simplifications as analyzed anddiscussed in Section III.

C. Simulation of Static Nonlinear Behavior and “Quasi-StaticMovement”

For calculating the deflection of a cantilever structure due toa constant external force distribution, basically only one singleiteration step as described in Section II-B would be necessary(linear quasi-static analysis). However, since the electrostaticforce depends on the deflection of the cantilever and is highlynonlinear, sequential iteration steps with slowly increasing de-flection have to be carried out until a steady state is reached(nonlinear quasi-static analysis). Especially when determining“static” parameters such as the pull-in voltage of an electro-static actuator, a quasi-static approach is sufficient and a full-dy-namic analysis is not necessary. Beyond that, the deflection ofsequential iteration steps mimics the dynamic movement of thecantilever. Thus, this algorithm is, for example, capable of pre-dicting the pull-in behavior of an electrostatically actuated can-tilever and can be used to determine the pull-in position or tosimulate the rolling behavior of a zipper-actuator. It should bekept in mind, though, that the presented quasi-static algorithm is


Fig. 4. Basic flowchart of the algorithm. The torques M(x ; t) are derived asshown in Fig. 3.

not thought to replace a full-dynamic analysis of dynamic pro-cesses, which includes the effects of kinetic energy and inertiaof the masses [24].

The basic flowchart of the algorithm is shown in Fig. 4. Forsimulating static nonlinear processes and “quasi-static move-ments” such as the pull-in behavior of electrostatic actuators,the algorithm is endowed with the following features.

• (Spatial) step-size adaptation for quasi-static modeling:The maximum change in the deflection between succes-sive iteration steps is limited to a parameter toensure the sufficient quantization of the highly nonlinearelectrostatic force.

• Contact detection: A contact is detected when the distancebetween the flexible structure and the isolation layer onthe rigid electrode gets below a certain limit. The positionand the slope of the contact point determine the constrainsand integration constants and for solving (5) in thesubsequent iteration. The algorithm is capable of detectingmultiple contact points in different iteration steps. Thus, itis capable of simulating the pull-in and rolling behavior ofa zipper-actuator.

Fig. 5. Demonstration of simulating “quasi-static” movements: Tip deflectionand dynamic step-size over the iteration steps of a cantilever pulling in alongthree stoppers. The cantilever shape of a few selected steps of the pull-in historyare displayed: A) critical pull-in position to the first stopper; B) contact with thefirst stopper; C) contact with the second stopper; D) critical pull-in position tothe third stopper; and E) contact with the third stopper and final steady state.(Actuator configuration: L = 400 �m, y = 2 �m, y = 20 �m,n = 2:5, n = 3, a = 1:2, a = 0:73, t = 4 �m, E = 150 GPa;simulation parameters: N = 2, (�y = 0:1 nm, n = 100, actuationvoltage = 104:2 V.).

• Detection of stable states: The algorithm terminates if thechange in deflection is below a critical limit for acertain number of subsequent iteration steps.

These features are demonstrated in Fig. 5, showing the simula-tion history of a cantilever pulling in along a curved-electrodeactuator, by displaying the cantilever deflection at a few selectedquasi-static iteration steps.

The pull-in voltage of an electrostatic actuator can be deter-mined by running a simulation series with different actuationvoltages. The pull-in position can easily be detected by iden-tifying the abrupt step in the plot of the final steady-state tipdeflection over the actuation voltage. A fast algorithm to accu-rately determine the pull-in voltage can be built around the algo-rithm presented in this paper by simulating a small set of actu-ation voltages whose values is refined after every iteration stepaccording to the threshold voltage found in the previous step.Fig. 6 shows how such an algorithm was used to determine thepull-in voltages of a cantilever successively snapping in to threestoppers when the actuation voltage is gradually increased.

III. INACCURACIES AND LIMITATIONS OF THE ALGORITHM

A. Applicable Class of Problems

Because this algorithm is specially written for a certain classof problems, it is only applicable for the following categories ofgeometrical problems:

• single-sided clamped electrostatic actuators;• two-dimensional geometry, i.e., homogeneous in the third

dimension;


Fig. 6. Determination of the pull-in voltages of a cantilever successively snap-ping-in on three stoppers: A) actuation voltage just below the pull-in voltage forsnapping in to the first stopper; B) and C) before pulling-in to the second andthird stopper; and D) final deflection.

Fig. 7. Nonlinearity error of the Euler–Bernoulli beam equation based algo-rithm as compared to FEM analysis. The error for a cantilever tip deflection upto 10% of the cantilever’s length is less than 1.5% and nearly independent of thecantilever thickness. (Configuration: L = 300 �m, E = 150 GPa, deflectiondue to a variable force at the tip).

• thin moving films/cantilevers (as compared to the length);• small deflections (as compared to the length).

These conditions, however, are typical for most MEMS electro-static actuators, which consist of thin films and are characterizedby small deflections. For these type of structures, the algorithmsupports complex geometries of the flexible and moving elec-trodes, including stoppers, isolation layers, thickness inhomo-geneities, and composite multilayer cantilevers.

B. Inaccuracies of the Algorithm

According to the limitations listed in Section III-A, the in-accuracies of the simulation results of the presented algorithmhave been investigated. Fig. 7 shows the nonlinearity error ofthe algorithm for large tip deflections. The nonlinearity errorfor a typical MEMS actuator with a deflection of 10% of thecantilever’s length is less than 1.5% and nearly independent ofthe cantilever thickness. A FEM model with large deflection el-ements and deformable mesh, simulated in COMSOL Multi-Physics, has been used as the reference model.

Fig. 8. Simulation error of the pull-in voltage of a mass-spring electrostaticactuator as compared to a full-dynamic model: (a) deflection over the itera-tion steps for the quasi-static model; (b) deflection over the time of a full-dy-namic model set-up in Simulink, with Q = 10; and (c) comparison of thepull-in voltages of the two models. Only for undercritical damping, the devi-ation of the quasi-static model is significant. Model data: mass-spring system ofa silicon cantilever; length, width, thickness = 100; 60;4 �m; spring constant= 10 Nm , resulting in mass = 48 ng, resonance frequency= 72:6 kHz; ac-tuation voltage in (a) and (b) is 50 V.

The algorithm is based on a quasi-static approach neglectingthe effects of inertia of the moving masses. Kinetic energy, how-ever, might even have an influence on apparently “static” pa-rameters, such as the pull-in voltage. Fig. 8(a) shows the deflec-tion of a cantilever mass-spring system over the iteration steps,modeled by the quasi-static approach for an actuation voltagebelow the pull-in threshold. Fig. 8(b) displays the time responseof a full dynamic model, clearly showing oscillating behaviorfor an undercritical damped system. Due to the oscillation, thecantilever in the full-dynamic simulation gets much closer to theelectrode than the final steady-state deflection in the quasi-staticapproach, which results in a lower pull-in voltage. Fig. 8(c) plotsthe error of the pull-in voltage of the quasi-static model overthe quality factor of the system. For a critical or overcriticaldamped system, the error of the quasi-static approach is less than0.5%. For an undercritical damped system, the error reaches upto 8.5% for a of 1000.

When comparing the performance of this quasi-static algo-rithm with a full-dynamic analysis, it has to be considered thatif the rise-time of the actuation voltage is larger than the rise-time of the system’s step response, the dynamic behavior ofthe system is fully equivalent to the quasi-statically simulatedbehavior. This point is quite relevant from a practical perspec-tive, since most MEMS electrostatic actuators have rather shortsystem response times. The rise-time of the step-response ofthe typical cantilever curved-electrode actuator investigated inFig. 8, for example, is only 4.82 s long.

C. Choice of Suitable Algorithm Parameters

The most important parameters of the algorithm are:• number of nodes ( -axis discretization);• maximum step-size per iteration step;


Fig. 9. Investigation of the influence of the parameters of the stability criterionon the accuracy of the pull-in voltage.

Fig. 10. Simulation error off the pull-in voltage depending on the x-axis dis-cretization of the cantilever (number of nodes and number of discrete capacitors)for different stability criteria parameter sets: (a) N = 100, (�y =1 nm; (b) N = 10, (�y = 1 nm; (c) N = 10, (�y =10 nm; and (d) N = 10, (�y = 100 nm. (Actuator configuration:L = 400 �m, y = 2 �m, y = 20 �m, n = 2:5, t = 3 �m,E = 150 GPa.)

• the two parameter of the stability criterion: numberof successive iteration steps with step-size smaller than

to determine a stable state.Fig. 9 displays the dependency of the simulated pull-in

voltage on the two parameters of the stability criterion for atypical MEMS curved-electrode actuator. For a threshold level

nm, the error of the pull-in voltage is less than0.1%, nearly independent of the second parameter . Evenwith an extremely coarse parameter set of nmand , the error created by the stability criterion isstill less than 1%.

The influence of the -axis discretization ( ) is illustratedin Fig. 10. For suitable parameter sets (a), (b), and (c) of thestability criterion, the error due to the -axis quantization is lessthan 1% for a number of nodes as low as 30, and is less than0.25% for more than 100 nodes.

Also the step-size limit has been investigated withthe results plotted in Fig. 11. Interestingly, for simulating thepull-in voltage or the steady state of a not pulling-in cantilever,the step-size limit has basically no influence on the simulation

Fig. 11. Dynamic, spatial step size over the iteration steps of a cantileverpulling in, for an actuation voltage just above the pull-in voltage. The step sizereaches a minimum at the critical pull-in point, which implies that the pull-involtage is independent on the maximum step-size limit (�y) . (Actuatorconfiguration: L = 400 �m, y = 2 �m, y = 20 �m, n = 2:5,t = 3 �m, E = 150 GPa, V = 51 V, V = 50:323V.)

results, since the dynamic step-size reaches a minimum aroundthe critical pull-in point or the final stable point, as shown in thefigure.

IV. PRACTICAL PERFORMANCE EVALUATION

A. Comparison to Other Simulation Methods

Besides the evaluation in Section III, the performance of thealgorithm in terms of predicting the pull-in voltage of typicalmicromechanical structures was compared to the following.

• Implicit, Rayleigh–Ritz method based simplified analyticaldescriptions of lateral curved-electrode structures withelectrode shape of different order as reported in the litera-ture [1]. The electrode shape is described by

(7)

with the cantilever length of 500 m, the initialelectrode gap of 30 m, the maximum electrodegap of 30 m, and the order of the electrode shape, whichis varied from 0 to 2.0. Further simulation parameters are

GPa, cantilever width 2 m, and cantilever thick-ness 5 m. The results obtained by the presented algorithmdeviate from the results of the simplified analytical modelbetween 0.5% and 7.5%, as summarized in Fig. 12.

• FEM simulations, a very simplified analytical model, andmeasurements of a laterally moving multilayer MEMSswitch, as shown in Fig. 13(a), etched in the device layerof a silicon-on-insulator wafer and sputter-coated with analuminum layer of a measured thickness of about 600 nm,fabricated at Nanyang Technological University, Singa-pore [25]. All of the simulations and calculations includingthe FEM analysis are based on the geometry model shownin Fig. 13(b), which is simplified at the thicker end ofthe cantilever as compared to the device structure. AFEM analysis reveals that this part of the cantilever of thereal structure is by a factor of 1.8 times stiffer than thecorresponding part of the simplified geometrical model


Fig. 12. Simulation of laterally moving curved electrode actuators with varia-tion of the electrode shape ordern, as compared to a simplified analytical modelbased on the Rayligh–Ritz method [1]. n = 0 is the special case of initially par-allel electrodes, and orders of n < 1 are not of any practical significance.

Fig. 13(d); however, since the main bending occurs at theinitial part of the cantilever, the inaccuracy in predictingthe pull-in voltage when using the simplified geometry(c) as compared to the the full geometry (b) was foundto be less than 0.13%. Thus, this simplification, whichwas necessary for the analytical analysis, is justified. Themultilayer Al/Si/Al structure is modeled for the analyticalanalysis by weightening the Young’s Modulus with thethicknesses of the different layers [25], whereas the multi-layer model of the presented algorithm, based on equation(1) is more correct. The measurement and simulationresults are summarized in Table I and reveal a very goodagreement between the different simulation methods andthe measured pull-in voltage of the fabricated devices.

• Measurements of a vertically moving, prestressed thin-filmzipper actuator fabricated for a MEMS switch at KTH[Fig. 14(a)]: the prestress bending was modeled by usingthe measured total stress gradient of 39 MPa m ofthe 1- m-thick silicon nitride membrane coated with a190-nm-thick Cr/Au layer [6]. The pull-in voltage of thefilm was determined by the presented algorithm to be49.6 V as compared to the measured value of 55 V (devi-ation of 9.8%). Also, the algorithm was used to predictthe dynamic behavior of the film pulling-in and rollingover the substrate electrode, as shown in Fig. 14(b). Theetched pattern in the fabricated structure is not taken intoaccount by the two-dimensional algorithm which assumesa homogeneous film; however, the structure in the 3rddimension does not affect the actuator’s behavior sincethe membrane areas coated by the electrode metal areof rectangular shape and fully congruent to their counterelectrodes on the substrate.

B. Performance Evaluation as a Design Optimization Tool

A suitable design optimization tool is supposed to predict thestatic and dynamic behavior of the actuator and to allow for the

Fig. 13. Laterally moving MEMS switch fabricated at NTU in Singapore:(a) scanning electron micrograph (SEM) picture; (b) schematic drawing of thesimulation model as used for the FEM simulations, the analytical model andthe algorithm presented in this paper; (c) FEM simulation of the deflection of amore accurate geometrical model of the thicker end of the beam; and (d) FEMsimulation of the same part of the cantilever for the simplified geometricalmodel from (b). The real geometry (c) is by a factor of 1.8 times stiffer than thesimplified geometry (d). However, since the main bending occurs in the initialpart of the cantilever, the inaccuracy in predicting the pull-in voltage whenusing the cantilever tip geometry (c) over the geometry (b) is less than 0.13%.

TABLE ICOMPARISON OF THE PULL-IN VOLTAGES OF THE SWITCH SHOWN IN FIG. 13

optimization of the actuator parameters, such as the geometryand the actuation voltage.

Analytical descriptions for touch-mode or curved-electrodeactuators are not available and FEM/BEM or FCM/BCManalysis, having the capability of dynamic simulations ofvery complex structures, typically take minutes to hours forsolving a highly nonlinear multicontact problem with complexgeometry, even for a two-dimensional model. Thus, they arenot practicable for use as a fast design tool for multiparameteroptimization.

The algorithm presented in this paper takes only a fractionof a second to solve a model as shown in the Figs. 5, 13,or 14, even when implemented in the slow script-languageof MATLAB on a standard personal computer with a clockfrequency of 1.3 GHz.


Fig. 14. Zipper actuator fabricated at KTH as part of a radio-frequency MEMS switch: (a) SEM picture and (b) selected quasi-static simulation steps of the pull-inand rolling movement of the KTH zipper actuator.

Fig. 15. Two-parameter variation for optimizing the geometry of a curved electrode to achieve maximum deflection of a 2-�m-thick 300-�m-long cantilever atan actuation voltage of 50 V. Because of the fast algorithm, the design optimization is based on a brute-force method, which means that all possible parametercombinations are simulated sequentially. The simulation results of four parameter sets A)–D) are plotted in Fig. 16.

The capabilities of the algorithm as a design optimizationtool are demonstrated on an example of a two-parameter ge-ometry optimization. The shape of a curved electrode was opti-mized to achieve the maximum deflection of a 2- m-thick and300- m-long silicon cantilever at an actuation voltage of 50 V.The electrode is covered with a 0.2- m-thick isolation layerand the minimum gap between the electrodes is 2 m. Equa-tion (7) describes the shape of the electrode with and

the parameters to be optimized. The geometry was optimizedby a brute-force method, i.e., the two parameters were varied by15 values each, and a total of 225 possible parameter combina-tions had to be solved. Because of the fast algorithm, the overallsimulation time took no more than a few seconds on a standard

personal computer. The achieved deflections for each parametercombination are plotted in Fig. 15, and the final stable states offour parameter sets (A)–(D) from Fig. 15 are plotted in Fig. 16 todemonstrate the different cases of stable end positions. Set (A)does not pull-in because the electrode shape order is too low,which results in a large initial curvature of the electrode gener-ating too little force on the cantilever. Set (B) uses an electrodeshape of higher order and the cantilever snaps in. However, thetip does not reach its maximum deflection since the electrodegap is increasing too fast at a larger distance from the fixedend of the cantilever. Finally, parameter set (C) achieves thelargest tip deflection of 21.56 m at an electrode shape order of

and a maximum electrode gap of m.


Fig. 16. Electrode shape and simulated final deflection of the curved-electrode actuator described by the four parameter sets A)–D) from Fig. 15: A) electrodeshape order too low, no pull-in; B) electrode shape order high, resulting in pull-in, but curvature of the electrode too large for a complete pull-in of the cantilevertip; C) optimal parameter set; D) larger maximum electrode gap at the optimum electrode order prevents pull-in.

The attempt of further increasing the deflection by a larger max-imum electrode gap does not anymore result in a pull-in,as demonstrated by set (D) in Fig. 16.

Especially for a multiparameter optimization, it is recom-mended to use an optimization algorithm based on the mini-mization of a function evaluating the simulation results of eachparameter set instead of the brute-force technique.

V. CONCLUSION

The presented algorithm is capable of predicting the staticbehavior and quasi-static movement of single-side clampedelectrostatically actuated MEMS structures. It can solve highlynonlinear and multiple-contact problems of geometries withcomplex elements including stoppers, isolation layers, andmultilayer structures. Thus, it is very suitable for simulatingcurved-electrode and touch-mode actuators. The accuracy andlimitations of the algorithm have been investigated. The algo-rithm’s performance has been compared to other simulationmethods and to measurements of fabricated devices, for whichthe algorithm was found to have good agreement. Because ofits efficient mathematical solver, it is faster by a few orders ofmagnitudes as compared to FEM-based software, which makesit a very fast, versatile, and powerful design tool for parameterevaluation and optimization of electrostatic actuator geome-tries. The algorithm’s main application field is seen in the fastdetermination of suitable parameter sets for an actuator, but it

cannot substitute a more accurate FEM analysis to investigatea final design in great detail.

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Joachim Oberhammer (M’07) was born in Italyin 1976. He received the M.Sc. degree in electricalengineering from the University of TechnologyGraz, Austria, in 2000 and the Ph.D. degree from theRoyal Institute of Technology, Stockholm, Sweden,in 2004 for his work in RF MEMS and wafer-scalemicrosystem packaging.

He was working with automotive sensor elec-tronics and RFID systems at the Graz Universityof Technology and Vienna University of Tech-nology, Austria, before he joined the Microsystem

Technology Laboratory at KTH-Royal Institute of Technology in Stockholm,Sweden. After having been a Postdoctoral Research Fellow at NanyangTechnological University, Singapore, he returned to the Royal Institute ofTechnology in 2005, where he currently an Assistant Professor and is heading aresearch team with activities in RF MEMS. He is author and coauthor of morethan 35 reviewed research papers and holds three patents.

Dr. Oberhammer received the Ericsson Research Foundation award in 2004.

A.-Q. Liu received the B.Eng. degree from Xi’anJiaotong University, China, in 1982, the M.Sc.degree in applied physics from Beijing Universityof Posts and Telecommunications, China, in 1988,and the Ph.D. degree from the National Universityof Singapore (NUS), Singapore, in 1994.

Currently, he is an Associate Professor at Schoolof Electrical & Electronic Engineering, NanyangTechnological University (NTU). He is also as anAssociate Editor for the IEEE SENSOR JOURNAL anda Guest Editor for Sensors & Actuators A: Physics.

His research interests are MEMS design, simulation, and fabrication processes.

Göran Stemme (F’05) was born in Stockholm,Sweden, in 1958. He received the M.Sc. degreein electrical engineering and the Ph.D. degree insolid-state electronics from Chalmers Universityof Technology, Gothenburg, Sweden, in 1981 and1987, respectively.

In 1981, he joined the Department of Solid StateElectronics, Chalmers University of Technology,where in 1990, he became an Associate Professor(Docent) heading the Silicon Sensor ResearchGroup. In 1991, he was appointed Professor with

The Royal Institute of Technology, Stockholm, where he heads the Mi-crosystem Technology Group, School of Electrical Engineering. His researchis devoted to microsystem technology based on micromachining of silicon. Theworks spans over a broad range of technological and application fields suchas medical technology, biochemistry, biotechnology, microfluidics, opticalapplications, wafer-level packaging and device integration. Some of the resultshave successfully been commercialized. Between 1995 and 2001, he was amember of the International Steering Committee of the Conference series IEEEMicroelectromechanical Systems (MEMS) and he was General Co-Chair ofthat conference in 1998. He has published more than 150 research journal andconference papers and has more than 12 patents proposals or granted patents.

Dr. Stemme is a member of the Editorial Board of the IEEE/ASME JOURNAL

OF MICROELECTROMECHNICAL SYSTEMS since 1997 and was a member of theEditorial Board of the Royal Society of Chemistry journal Lab On A Chip be-tween 2000 and 2005. In 2001, he won (together with two colleagues) the finalof Innovation Cup in Sweden. He is a member of the Royal Swedish Academyof Sciences (KVA).

Dr. Stemme is a member of the Royal Swedish Academy of Sciences.Between 1995 and 2001, he was a Member of the International SteeringCommittee of the IEEE Microelectromechanical Systems conference and wasGeneral Cochair of that conference in 1998. He has been a Member of theEditorial Board of the IEEE/ASME JOURNAL OF MICROELECTROMECHNICAL

SYSTEMS since 1997. In 2001, he received (with two colleagues) the InnovationCup in Sweden.

Documents

Time-Efficient Quasi-Static Algorithm for Simulation of Complex Single-Sided Clamped Electrostatic Actuators